Problem: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $26.5$ years; the standard deviation is $2.3$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living less than $31.1$ years.
Answer: $26.5$ $24.2$ $28.8$ $21.9$ $31.1$ $19.6$ $33.4$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $26.5$ years. We know the standard deviation is $2.3$ years, so one standard deviation below the mean is $24.2$ years and one standard deviation above the mean is $28.8$ years. Two standard deviations below the mean is $21.9$ years and two standard deviations above the mean is $31.1$ years. Three standard deviations below the mean is $19.6$ years and three standard deviations above the mean is $33.4$ years. We are interested in the probability of a snake living less than $31.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $21.9$ years and the other half $({2.5\%})$ will live longer than $31.1$ years. The probability of a particular snake living less than $31.1$ years is ${95\%} + {2.5\%}$, or $97.5\%$.